A tangent line is a line that only passes through one point of a function and represents the slope at that point on the function. A normal line is the line that is perpendicular to the tangent line of the function.
Ex: If the slope of the tangent line is 3/4, then the slope of the normal line would be -4/3.

http://t1.gstatic.com/images?q=tbn:ANd9GcRNPnU3ZTdkSCHq22MhrY0suD52jkM279vKEQVRCczk8Ldr7ekMsg